The "fifth straight" losing season means that the team had a losing season 5 seasons in a row. In other words, they had five losing seasons, one after another, with no winning seasons in between.
This phrase is very useful for talking about numbers and records which are interesting and unusual. It's most common for sports- and economics-related discussions.
This is their third straight national championship.
The stock market fell for the 20th straight day today.
When you're talking about something that's no longer current, use "___ straight" instead of "the ___th straight":
They've had four straight losing seasons, but they're hoping that this year will be different.